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A block (mass $m_{A} )$ lying on a fixed frictionless inclined plane isconnected to a mass $m_{B}$ by a cord passing over a pulley, asshown in Fig. $54 .(a)$ Determine a formula for the accelera-tion of the system in terms of $m_{A}, m_{B}, \theta,$ and $g .$ (b) What conditions apply to masses $m_{\mathrm{A}}$ and $m_{\mathrm{B}}$ for the accelerationto be in one direction (say, $m_{\mathrm{A}}$ down the plane), or in theopposite direction? Ignore the mass of the cord and pulley.

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a. $g \frac{\left(m_{\mathrm{A}} \sin \theta-m_{\mathrm{B}}\right)}{\left(m_{\mathrm{A}}+m_{\mathrm{B}}\right)}$b. $m_{\mathrm{A}} \sin \theta>m_{\mathrm{B}}(\text { down the plane })$, $m_{\Lambda} \sin \theta<m_{\mathrm{B}}(\text { up the plane })$

Physics 101 Mechanics

Chapter 4

Dynamics: Newton's Laws of Motion

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Applying Newton's Laws

Moment, Impulse, and Collisions

Rutgers, The State University of New Jersey

University of Washington

Hope College

Lectures

03:28

Newton's Laws of Motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows: In his 1687 "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), Isaac Newton set out three laws of motion. The first law defines the force F, the second law defines the mass m, and the third law defines the acceleration a. The first law states that if the net force acting upon a body is zero, its velocity will not change; the second law states that the acceleration of a body is proportional to the net force acting upon it, and the third law states that for every action there is an equal and opposite reaction.

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

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So here, uh, for part A, we need to, of course, drama free body diagram for each block so we can draw the I'm. So be first going up would be forced. Tension going down would be mass of block B times G. So essentially the weight going up would be the displacement in the wind direction for block B. So we're gonna say upwards is positive. Uh, for the second clock, we have perpendicular to the surface of contact. Forced, normal. Going straight down would be m sub G or rather, M c a g. This would be our angle theta, uh, and then going straight to the left, this would be forced tension. And so we can say that the sum of forces in the UAE direction for block a would be force normal minus M sub a g co sign of data this would equal zero. We can say forced normal would then be equal to m sub a g coastline of Fada. We can say that some of forces in the ex direction for Block A would be equal to M. C. A g sign of fada and then minus force of tension. This would equal the mass of a times the acceleration of block A in the ex direction. We can say that the sum of forces, uh, in the UAE direction for Block B would be equal to force tension minus m sub B g. This is equaling m sub b times acceleration of block B in the right direction and therefore forced tension would be equal to M sub B times jeep, plus the acceleration in the wider action of Block B. And so we can say that since the blocks are connected by a cord, the acceleration of block B in the right direction would be equal to the acceleration of luck, eh? In the extraction and this would simply be equal to a the acceleration. Let's substitute the expression for the tension force into the last equation into the from Sorry, let's substitute this equation rather into the equation for block One. So we can say that EMS of a G, a sign of fada minus mass of B times G plus a would be equal to mass of eight times a ah therefore massive a G sign of data minus massive B A G would be equal to mass of a times a plus mass of B times A. And we find that the acceleration is gonna be able to g times mass of a sign. If ADA minus massive block be divided by the total mass and a plus B, this would be your formula for acceleration for part A now, for part B. However, if the acceleration is to be down the plane, it will be positive. So if hey is down the plane Hey was created in zero and this would happen if m sub a sign of Fada is greater than m sabi. Now if it's up the plane, if acceleration is up the plane Uh, this means that the acceleration is less than zero and m sub. A sign of Fada is going to be less than NASA. Be again. This would be for up the plane. This would be for down the plane if and send a sign of Fada equals m ster B. Then we know that the acceleration is gonna equal zero meters per second squared and the entire system will either be at moving at a constant velocity or be completely at rest. That is the end of the solution. Thank you for watching

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